In mathematics education both modern and postmodern critical research focus on what could be otherwise, wanting to investigate the consequences of implementing differences. This paper discusses the similarities and the differences between the two approaches. Both approaches see a borderline between necessity, to be accepted, and eventuality, to be criticised and possibly changed. Modern critical thinking takes both education and mathematics for given, and only considers educational methods as objects for critique and change. Postmodern critical thinking also includes both education and mathematics, suggesting them to be rephrased as ‘seducation’ and ‘grammar of our number-language’ to make visible the political nature of these ruling phrasings, both enabling and constraining our actions.
Modern critical thinking has its roots in Frankfurt and Habermas. It believes in emancipation from ruling structures, and goes to the barricades to substitute authoritarian will with democracy and power-free argumentation. It is critical towards the misuse of research, but accepts research as a prerequisite for democracy.
Postmodern critical thinking has its roots in Paris and Foucault. It believes in emancipation from ruling discourses, and goes into the text to substitute ruling echo-phrasings with counter-phrasings, seeing the ‘democratic power-free argumentation’ as rather a power game between competing wills to knowledge. It is critical towards research, seeing research as interpretations in disguise constraining democracy by presenting research results as information and not as informed opinions for debate.
The dark Middle Age fell upon Europe when the Arabs conquered the silver mines in Spain and North Africa. The social order became a double feudalisation with two rulers, a concrete and an abstract, a physical and a metaphysical, the King and the Creator. The darkness turns into twilight when silver was found in Germany and discovered in America, making world trade include Europe again. England wanted to participate, but had to sail on open sea to India to avoid the Portuguese. Hence knowledge of the motion of the moon became a crucial question to the English. Newton found the answer by showing that falling bodies as apples and moons are obeying a metaphysical law. Not the law of the Creator, which is unpredictable according to the church, so all we can do is to believe and prey. But a law of motion, that is predictable by being formulated in numbers and calculations, so all we have to do is to know and calculate. Newton’s discovery replaces Christianity with Platonism in the metaphysical part of the feudalisation, and creates the foundation for enlightenment, industrialisation and democracy, in short for modernity. As to the physical part, the king, Kant argues for a new humanistic enlightenment by asking "Was ist Aufklärung (what is enlightenment)?" and answering:
Enlightenment is man's emergence from his self-incurred immaturity. Immaturity is the inability to use one's own understanding without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. The motto of enlightenment is therefore: Sapere aude! Have courage to use your own understanding! … For enlightenment of this kind, all that is needed is freedom. And the freedom in question is the most innocuous form of all – freedom to make public use of one’s reason in all matters. (Kant 1784)
Kant thus replaces the master/servant relationship with a political public space populated by free, equal and empowered humans - humans can become their own masters. However feudal masters do not give up that easy. They have to be removed in one of two ways, by escape or by revolution. With its "freedom under Good" principle the US removes the double feudalisation with a stroke of the pen. On this background John Dewey formulates in the beginning of the 20th century a pragmatism linking democracy and education together (Dewey 1916). Germany had to go through revolutions and wars. After the First World War the new German democracy was sabotaged by the treaty of Versailles putting a drainage tube into the German economy (Keynes 1920). After the Second World War the physical part of the double feudalisation seemed finally removed, so after a period of rebuilding the enlightenment dream of empowerment now finally could be implemented. The Frankfurter school formulates a democratic critical enlightenment ideal, where Adorno puts as the first claim to a modern school, that it prevents a new Auschwitz (Adorno 1988). As a road to democracy Habermas proposes the power free dialogue where the most convincing argument wins (Habermas 1996). Also in the Danish enlightenment debate the Frankfurter school inspires to a concept of critical enlightenment:
Besides specific knowledge enlightenment also contains a criteria for the application of this knowledge accepting responsibility for how, when and for what the knowledge is applied (Nielsen 1973 p. 40-41).
The critical enlightenment ideal is not an alternative but an extension of the traditional ideal thus importing its metaphysical Platonic enlightenment ideal. This ideal is however problematised by the emergence of postmodernity and postmodernism on the other side of the Rhine. The power free dialogue of Habermas could also be considered a power game between competing phrasings and discourses trying to feudalise and clientify the other (Foucault 1972). And Bauman points out how three central aspects of modernity: authorisation, routinisation and dehumanising can hide amorality in a web of transfer of morality, technical rationality and covering phrasings. Auschwitz might not be an abnormality but a normality of modernity (Bauman 1989).
So the transition period between modernity and postmodernity sees two forms of critical thinking, a modern and a postmodern version.
In the paper ‘Research Methodology and critical Mathematics Education’ (Skovsmose et al. 2000) Ole Skovsmose and Marcelo Borba argue that curriculum research should be organised in a triangle of situations: the current, the imagined and the arranged. And its quality should be evaluated in terms of pedagogical imagination, practical organisation and critical reasoning. Finally change in education should be described as a trajectory of triangles. To exemplify this, an example is included in their paper:
In Brazil there is a research group, which has focused on issues related to new technologies and mathematics education. This research group has developed software and work with students at different levels and with teachers. A teacher from a nearby school approached the group through her supervisor manifesting her interest in using computers with her students. Her interest was vague, and one of the members of the research group decided to work with her. From the standpoint of the researcher, there was the interest in developing investigation about how a teacher, who did not know much about computers, would appropriate this new technology and the interest in supporting the teacher. From the teacher perspective, she had some tough problems to face and she foresaw that the computers would be able to help her. The teacher was teaching a class of 5th graders, which in her view was really problematic. The kids were older (15 years average) than the expected age for this grade: 11. The kids felt humiliated somehow as they were put in a school with kids much younger than them and they had flunked many times, and at several instances they had to repeat all the subjects of a given school year because their ‘failure in mathematics’. The students transformed this humiliation into violence in class. The teacher was in fact considering the possibility of just quitting the job since she could not work with those kids in a way she found effective.
The teacher saw the computer as a way of trying to engage the students in mathematical thinking. The teacher and researcher started meeting regularly so that they could agree on pedagogical issues regarding the use of computers. In these meetings there were teaching and learning from both sides. There was a big problem, though: the computers, which had been promised by the state government, had not arrived at the time when teacher and researcher had agreed to start using them in the classroom and the school had only four computers which were in a room that would only hold eight people. There were no computer facilities, therefore, for 25 students in the school. The researcher suggested to the teacher that she might use the laboratory of the university for the lessons in which she wanted to use the computer. The teacher enjoyed the idea and believed that students could easily walk to the lab.
The teacher was enthusiastic about a software, which deals with rational numbers. The teacher took the lead in the choice of the program and in the design of the tasks, which were presented, to the students. Although the research group believed that this software is full of root learning tasks, they agreed to use the software. This may ease the transition of the teacher into a classroom with computers. Besides, both researchers and teacher had the ‘intuition’ that the computer might have a positive effect in this class and maybe could avoid that the students had to repeat this grade again. (p. 6-7)
According to Skovsmose and Borba they have not attempted “to choose an example which is exemplary of research in critical mathematics education. The purpose of introducing the example is to illustrate concerns within this kind of research and to use this example for a discussion of qualities of research which is concerned with changes in the classroom.” (p. 6) The example shows the different parts of the critical research paradigm. The three situations: a current situation CS with a teacher in a violent class, an imagined situation IS with computers at the school with flexible software “suitable to open-ended tasks and activities which can provide new insights in mathematics” (p. 14), and an arranged situation AS with computers at the university with inflexible software “full of root learning tasks” (p. 7).
Also there are dimensions of quality. The pedagogical imagination PI relates the current and the imagined situation. Both the teacher and the researchers had a vision about computers bringing benefits to the classroom making them choose to cooperate:
In the example we can argue that within the research group there is a strong ‘utopia’ about computers being able to support students in their own investigation. ... Such ideas were here negotiated with the teacher, ... (p. 14)
The practical organisation PO relates the current and the arranged situation:
We find this quality to be closely linked to the quality of the cooperation between teachers, researcher, students and administrators. To organise an arranged situation means to negotiate a specific situation with specific constraints. For instance, it was possible to use the laboratory of the university only because all parts involved agreed upon it. If there were no consensus, there were no way the arranged situation could happen. (p. 15-16)
The critical reasoning CR relates the imagined and the arranged situation. This is not illustrated by the example, but in terms of ‘expanding data’, a process, which has been suggested by Morten Blomhøj (1998):
The ‘data’, which are analysed, are not the observed data. Direct observations concerning communication in the classroom might be presented in forms of transcripts. But it makes sense to ask what would happened if, say, the teacher asked the students a different question and the communication thus took a different route from what is seen in the transcript. It makes sense to ask what would happen if students, working in a dynamic geometry environment, drag a different point of an object than the one they actually did. The not-experienced alternatives can be specified and presented in the form of an invented dialogue. (p. 17)
Finally the authors discuss the notion of transformation refusing to accept the following description of critical research: “The inquirer is cast in the role of instigator and facilitator, implying that the inquirer understands a priori what transformations are needed.” (Guba and Lincoln, 1994, p. 113)
To put the understanding of what transformations are needed in the hand of the researchers easily comes to represent dogmatism in the form of we-know-better. What had to be done is not possible to identify a priori. (p. 21)
Working with exemplary examples is a must in critical thinking. Why was it not possible for the authors to choose an exemplary example? Is it because the method is too new, or too difficult to apply in real life?
The authors stress the point, that nobody has access to unquestionable knowledge, and they refuse Guba and Lincoln’s characteristic of doing critical research (p.15). Again and again they stress the importance of negotiation and cooperation between students, teachers and researchers. Yet in the example students are not involved in the negotiations, only the teacher. And as to the pedagogical imagination it is the research group, that 1) has a ‘strong utopia’, which then is negotiated with the teacher, and 2) has the resources to arrange the practical situation, and 3) has the opinion, that flexible software is better than root learning software, which is temporarily accepted to help, not the pedagogical process, but the feelings of the teachers and the students. So in the example the research group seems to know what has to be done a priori, and to be instigators by focusing on computers in education before being asked by a specific classroom situation to do so.
The authors emphasise “that in order to do critical research it does not make sense to turn people involved in the research projects as teachers and students into ‘research objects’” (p. 19). Still the researcher in the example is making the teacher a ‘research object’ by “developing investigation about how a teacher, who did not know much about computers, would appropriate this new technology” (p. 6).
Finally the paper does not report on the outcome of the experiment in spite of talking about development trajectories (p.19). What happened to the class and the teacher next year? Did the students finally pass? Was the teacher satisfied etc.? Did the research group change their pedagogical imagination from the experience?
Also the third dimension of the quality is not illustrated from the example but from a different example, where a researcher is inventing data and narratives to report what happens in the classroom. Does this researcher have a solid classroom and teaching background to know what can happen in a classroom, or are the invented narratives pure speculations and fantasies? Are the invented narratives negotiated with the students and the teachers?
So even if critical research can be described as in the paper, it seems somewhat difficult to implement, raising the classical Cinderella-question: Are there other forms of critical research?
To get an idea about the difference between modern and postmodern critical research I will retell the example above as a fiction assuming that research group consists of postmodern critical researchers.
The research group is working halftime in classrooms and halftime at the university. It focuses on the concerns of typical classrooms as expressed by students, teachers in their stories of complaints. The teacher complains about the violence in the class tempting her to quit the job since she cannot work in a way she finds effective. And the students complain about having to repeat the class because they don’t want to learn about fractions, since the teacher by just echoing the textbook is unable to explain to the students, why they shall learn fractions, and what they are useful for (Tarp 1998). Asked to comment this, the teacher says that mathematics education means education in mathematics, and since rational numbers is part of the mathematics textbook it must be taught and learned. Mathematics is difficult to learn, so the students have to work harder, or be supported by computers. Hence the problems will not disappear before schools can afford computers, or the students decide to become more engaged in mathematics.
Based upon the motto “echo-phrasing is freezing, re-phrasing is freeing” postmodern thinking sees modern institutions frozen in echo-phrasings, that have to be discovered and rephrased. Since the teacher is echoing the textbook, the echoes can be found here. The textbook presents fractions as examples of rational numbers, being example of number sets, being examples of sets. This is the typical way of presentation within modern set-based mathematics explaining concepts as examples of more abstract concepts. This phrasing conflicts with the student demand for explanations relating fractions to their use.
So instead of developing software to supplement, and thus support, the existing top-down echo-phrasing of fractions, the group begins to look for alternative bottom-up approaches in journals, other textbooks, other countries, and in other time periods. Also they use their imagination by accessing the silent part of their ‘knowledge-iceberg’ developed through years of classrooms experience as mathematics educators. Using curriculum architecture they design examples of micro curricula, where fractions emerges from dividing problems, that can be introduced into the ordinary classroom as e.g. games, where students work in pairs throwing dices and splitting the profit, or loss, proportional to their stakes shown by their dice-numbers.
This ‘proportional splitting’ approach leads to (and thus shows the authenticity and necessity of) fractions, and multiplication of fractions and integers. But it will not lead to addition of fractions, which the tradition introduces before multiplication. A closer analysis uncovers addition of fractions as ‘killer-mathematics’: you never meet it outside the classroom, and it can only be used to kill the interest of the students. Outside the classroom 1 coke out of 2 bottles and 2 cokes out of 3 bottles add up to 3 cokes out of 5 bottles, but inside the classroom 1/2 + 2/3 = 7/6, i.e. a total of 7 cokes out of 6 bottles - a meaningless answer that however can be learned as an echo to pass the exam. Adding fractions grows out of trade problems, where adding parts of wholes always produces a part of the wholes, and never more than the wholes: 1/2·2 + 2/3·3 = 3 = 3/5·5, and 1/2·4 + 2/3·3 = 4 = 4/7·7. So 1/2 and 2/3 can add up to 3/5 or 4/7 or other fractions depending on the wholes, but never to 7/6, as the school says.
This micro-curriculum gives the students what they want, explanations from below. It will however have to be negotiated with the teacher before the educational material is worked out and brought to the classroom to be tested, and in this process the researcher must be able to take over the chalk if needed to be a real and authentic participant in the testing. If the material settles on a stable form after being redesigned a number of times, it is time to publish the micro curricula so it will be available to other classrooms as well. And time to begin the slow and laborious work to argue for a change in the official curriculum.
Postmodern critical research has quality if it has produced a difference that makes a difference. The micro-curriculum is a difference. Its micro-quality is measured by the percentage of successful students now passing after having flunked before. And its macro-quality is measured by its degree of success in changing the official curriculum to become more student friendly.
In their paper Ole Skovsmose and Marcelo Borba follow the modern critical tradition that political alternatives should be negotiated between participants, i.e. between students, teachers and researchers:
In this paper we present a perspective on research in mathematics education, which acknowledges the political aspects of this education and sees curriculum development as a process of negotiation. ...The quality of participatory research can then be discussed ion terms of ... cooperation between students, teachers and researchers.” (p. 1)
By using the words ‘students, teachers and researchers’ the modern critical thinking accepts the traditional modern division of labour within knowledge: Knowledge is, and knowledge is searched by researchers, taught by teachers, and studied by students. So the modern knowledge institutions are taken for granted. Curriculum itself is accepted; only its development can be negotiated.
Postmodern critical thinking sees a different borderline between necessity and eventuality, between what cannot be changed and what can, based upon the ‘pencil-dilemma’: Placed between a ruler and a dictionary a pencil can show its length but not its name; thus numbering reflecting indisputable natural necessity (“its length is 8±1 cm”) can form the basis for research, while wording can only reflect disputable political correctness (“it is a pencil”) and thus only form the basis for interpretations. Thus production of indisputable statements, research, can take place within natural science numbering the five necessities of nature: extension in space & time, mass & charge and multiplicity. Whereas all word-statements are interpretations, becoming seductions if presented as research, making the word-part of modern education ‘seducation’ seducing students, and teachers and researchers, to believe it is a true representation of an outside reality, instead of a political correct interpretation.
Using word-statements modern critical research should be rephrased as critical interpretation. However Descartes’ “doubt is undoubtable” allows meta-statements to be indisputable even if all statements can be disputed. This opens op for a postmodern critical research, a counter-research, producing counter-interpretations to interpretations presented as research, by rephrasing the echo-phrasings of the interpretation, using discovery or invention.
According to postmodern counter-research words are two-faced. One the one side our words allow us to talk, think, discuss and decide upon actions, and thus to transform nature into culture and autocracy into democracy. But when phrasings as ‘knowledge’, ‘research’, ‘teacher’, ‘student’, ‘education’, ‘mathematics’ etc. turn into echo-phrasings that is never rephrased, then the echo-phrasing becomes the invisible autocrat of modern ‘democracy’ determining and constraining the way we can think and act. Foucault calls this the subjection of the modern human being, and he also shows how modern rational science phases (and thus constructs) human abnormalities as e.g. ‘uneducatedness’ (Foucault, 1972). Once ‘discovered’ the abnormality can be studied by modern research, and scientific knowledge can be generated to be applied in building modern treatment-institutions to cure the abnormality, thus producing progress and development.
So the dream of the modern educational, health and correction institutions is to cure the uneducated mind, the unhealthy body and the incorrect behaviour, i.e. to cure abnormality based upon scientific diagnoses. Reality however is cruel. Instead schools, hospitals and prisons in many cases produces illiteracy, sickness and crime, thus creating big problems for the modern institutions, making them ask modern research to help. But only able to study a reality constructed by itself, modern research produces ‘fireworks-research’, being beautiful and expensive, but leaving the night as dark and cold as before. Instead the late modern school tries to solve its problems by forcing upon the uneducated a responsibility for their own education, and by supplying them with individual portfolios to help the teacher differentiate the education.
From a postmodern perspective this ‘improvement’ can be rephrased to ‘a survival attempt transforming the school from a prison to a hospital, and transforming students from inmates to patients with their own personal case records’. The late modern school is not being improved, but dehumanised. And by using the vocabulary of modern education, critical research is supporting this dehumanisation.
From a postmodern perspective the educational institution should be discussed both in terms of the traditional vocabulary ‘student, teacher, researcher’, but also of a rephrased vocabulary ‘inmates, guards, judges’ and ‘patients, nurses, doctors’. Modern critical research wants to have cooperation and negotiation between students, teachers and researchers. Does it also want cooperation and negotiation between inmates, guards and judges? And between patients, nurses and doctors? And if the uneducated must accept a responsibility for their own education, does that mean that also the criminals must accept a responsibility for their own correction, and that disease carrier must accept a responsibility for their own cure? Or is it better to accept, that modern institutions are places of authority, and to try to make the authority authentic, real and rational and instead of artificial, false and inhibiting? (Fromm, 1941)
Education is more problematic than its sister institutions. At the hospital or at the prison you are turned into a client, only if you have received the individual judgement ‘ill’ or ‘criminal’ by a doctor or a judge. And the treatment varies with the different forms of abnormality. At the school everybody is automatically condemned ‘uneducated’ from birth. Hence education do not have to name the abnormality, it is supposed to cure, it is taken for granted, and only gets an identity as a lack of education, which automatically calls for education as a cure.
In this way education gets a self-referring application-rationale preventing it to discuss the form of the ‘uneducatedness’ it is supposed to cure: Education is a necessity since it is applied in all aspects of life. And of course education must be acquired before it can be applied, if not there is nothing to apply. So first you must be educated, then you can enter into life. This application rationale runs through all the subject of modern education, including mathematics.
Instead of discussing e.g. quantitative illiteracy, mathematics is taken for given, and only its application and education is discussed according to the rationale “mathematics is, and mathematics is applied, hence mathematics must be taught.” Within this seducing rationale mathematics education can only be instituted from above, top-down: “Of course mathematics must be learned before mathematics can be applied. If not, there is no mathematics to apply!”
The echo-phrasing mathematics might allow an institution to be created to cure quantitative illiteracy, but if the cure does not work, if problems start to arise within the modern institution (see e.g. Jensen et al., 1998), the modern research start to look for scapegoats among the human factors testing hypothesises as: It is the fault of the teachers, of the students, of the politicians, of the parents etc. Postmodern research instead says, that it is never the fault of the humans; it is the fault of the ruling echo-phrasings, which need to be rephrased.
Rephrasing can be discovered by searching in space and in time, or invented by using the principle of full or high necessity. Travelling in time we find that mathematics used to be called ‘reckoning’ and ‘rechnung’ in Denmark and Germany, and that a mathematician used to called a ‘geometer’ in France. Travelling in space we find that in Japan mathematics is called ‘sansuu’ and ‘suugaku’ meaning ‘calculating and studying numbers’. In all these phrasings, ‘numbers’ is a central concept.
Using the principle of full or high necessity we can say, that multiplicity is a natural necessity which has created numbers as a cultural necessity. And numbers can be combined in calculations thus forming a ‘number-language’, which can be studied from its meta-language. Thus mathematics can be rephrased as ‘a grammar of the-number-language’. Within this rationale mathematics education can only be instituted from below, bottom-up: “Of course the language must be learned before the grammar of the language. If not, there is no language for the grammar to describe!”
Rephrasing ‘mathematics’ to ‘a grammar of the-number-language’ is making a difference. But is it a difference, which makes a difference, so it can be counted as counter-research. That has to be tested in the classroom.
The modern critical thinking was influential at the democratic revolutions around 1970 helping the late modern society to move the borderline between necessity and eventuality, so that unpolitizised issues became politizised and object for democratic discussions and decisions. In today’s postmodern society the time has come to repeat this success and to move the borderline once more. In 1970 the institutions were democratised, and now it is time to rehumanise them by expanding democracy to also include their ruling discourses determining the way we can think and act within mathematics and education. Modern critical thinking however has problems leaving its modern perspectives and theorists. It needs a ‘post-modernisation’, which maintains the critical perspective, but changes the objects of critique from visible outside rulers to invisible inside rulers, and changes its theoretical foundation from Germany to France.
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